Accuracy of the WKB Approximation: The Case of General Quartic Potential

We analyse the accuracy of the approximate WKB quantization for the case ofgeneral one-dimensional quartic potential. In particular, we are interested inthe validity of semiclassically predicted energy eigenvalues when approachingthe limit $E\to \infty$, and in the accuracy of low lying energy levels belowthe potential barrier in the case of generally asymmetric double-well quarticpotential. In the latter case, using the standard WKB quantization an unnaturallocalization of eigenstates due to the negligence of tunneling is implied andthus the validity of semiclassics is uncertain. In all computations the higherorder corrections to the leading semiclassical approximation are included usingthe complex contour integration technique. We show that these corrections canimprove accuracy of semiclassical approximation greatly by many orders ofmagnitude.Comment: 20 pages, 8 figures, 2 tables, PTP LaTeX style, to be published in the proceedings of the conference/summer school 'Let's Face Chaos through Nonlinear Dynamics', Maribor, Slovenia, June/July 1999, eds. M. Robnik et al., Prog. Theor. Phys. Suppl. (Kyoto) 139 (2000